Two Main Proof Styles

There are two major ways to prove things in mathematics: directly or indirectly.

Direct Proofs

A direct proof starts from known facts and uses logical steps to get to the statement you want to prove. Think of it as a straight path from start to finish.

Indirect Proofs

Indirect proofs include two important styles:

• Proof by Contradiction: Assume the opposite of what you want to prove. If that leads to something impossible, your original statement must be true!
• Proof by Contrapositive: Prove that if the conclusion is false, then the premise must also be false.

Choosing the Right Approach

Sometimes one style is easier or more efficient than the other. The trick is picking the method that makes your proof clear and convincing.

Logical Connections

Both types rely on logic, but indirect proofs can feel like detective work—spotting clues and eliminating false leads.

Practice Makes Perfect

Learning to switch between proof methods is a sign of a strong mathematical thinker.